The Tenth British Mathematical Olympiad
1974年第十届英国奥林匹克数学竞赛 
 C is the curve y = 4x^{2}/3 for x ≥ 0 and C' is the curve y = 3x^{2}/8 for x ≥ 0. Find curve C" which lies between them such that for each point P on C" the area bounded by C, C" and a horizontal line through P equals the area bounded by C", C and a vertical line through P .
 S is the set of all 15 dominoes (m, n) with 1 ≤ m ≤ n ≤ 5. Each domino (m, n) may be reversed to (n, m). How many ways can S be partitioned into three sets of 5 dominoes, so that the dominoes in each set can be arranged in a closed chain: (a, b), (b, c), (c, d), (d, e), (e, a) ?
 Show that there is no convex polyhedron with all faces hexagons .
 A is the 16 x 16 matrix (a_{i,j}). a_{1,1} = a_{2,2} = ... = a_{16,16} = a_{16,1} = a_{16,2} = ... = a_{16,15} = 1 and all other entries are 1/2. Find A^{1 }.
 In a standard pack of cards every card is different and there are 13 cards in each of 4 suits. If the cards are divided randomly between 4 players, so that each gets 13 cards, what is the probability that each player gets cards of only one suit ?
 ABC is a triangle. P is equidistant from the lines CA and BC. The feet of the perpendiculars from P to CA and BC are at X and Y. The perpendicular from P to the line AB meets the line XY at Z. Show that the line CZ passes through the midpoint of AB.
 b and c are nonzero. x^{3} = bx + c has real roots α, β, γ. Find a condition which ensures that there are real p, q, r such that β = pα^{2} + qα + r, γ = pβ^{2} qβ+ r, α = pγ^{2} + qγ + r.
 p is an odd prime. The product (x + 1)(x + 2) ... (x + p  1) is expanded to give a_{p1}x^{p1} + ... + a_{1}x + a_{0}. Show that a_{p1} = 1, a_{p2} = p(p1)/2!, 2a_{p3} = p(p1)(p2)/3! + a_{p2}(p1)(p2)/2!, ... , (p2)a_{1} = p + a_{p2}(p1) + a_{p3}(p2) + ... + 3a_{2}, (p1)a_{0} = 1 + a_{p2} + ... + a_{1}. Show that a_{1}, a_{2}, ... , a_{p2} are divisible by p and (a_{0} + 1) is divisible by p. Show that for any integer x, (x+1)(x+2) ... (x+p1)  x^{p1} + 1 is divisible by p. Deduce Wilson's theorem that p divides (p1)! + 1 and Fermat's theorem that p divides x^{p1}  1 for x not a multiple of p.
 A uniform rod is attached by a frictionless joint to a horizontal table. At time zero it is almost vertical and starts to fall. How long does it take to reach the table? You may assume that ∫ cosec x dx = log tan x/2.
 A long solid right circular cone has uniform density, semivertical angle x and vertex V. All points except those whose distance from V lie in the range a to b are removed. The resulting solid has mass M. Show that the gravitational attraction of the solid on a point of unit mass at V is 3/2 GM(1 + cos x)/(a^{2} + ab + b^{2}).

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